Quantitative estimates on the Hydrogen ground state energy in non-relativistic QED

TL;DR

This paper rigorously calculates the hydrogen ground state energy in non-relativistic QED, revealing logarithmic corrections and non-analytic behavior of the binding energy as a function of the fine-structure constant.

Contribution

It provides the exact expression for the hydrogen binding energy up to a specific order and proves the non-analyticity and logarithmic corrections in the energy expansion.

Findings

Exact expression for hydrogen binding energy up to O(α^5 log α^{-1})

Proof of non-analytic dependence of energy on α

Verification of logarithmic corrections in the energy expansion

Abstract

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order $O(\alpha^5\log\alpha^{-1})$, where $\alpha$ denotes the finestructure constant, and prove rigorous bounds on the remainder term of the order $o(\alpha^5\log\alpha^{-1})$. As a consequence, we prove that the binding energy is not a real analytic function of $\alpha$, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of $\alpha$, as conjectured in the recent literature.